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According to an answer on this MO post, showing that $$\int_{0}^{\infty}\frac{(1-12t^2)}{(1+4t^2)^3}\int_{1/2}^{\infty}\log|\zeta(\sigma+it)|~d\sigma ~dt=\frac{\pi(3-\gamma)}{32}$$

$($$\gamma$ is the Euler-Mascheroni constant$)$ is equivalent to the Riemann hypothesis.

I have two questions:

$(1)$ Has any serious attempt been made to evaluate this numerically or determine strong bounds?

$(2)$ Would numerically evaluating this integral be a valid heuristic argument in favour of the Riemann hypothesis?

Certainly, no amount of numerical accuracy constitutes a proof. However, if we show the equality holds to, say, a quadrillion digits or something, it will be true for all sakes and purposes; I doubt any mathematician would then seriously deny the validity of the conjecture.

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    $\begingroup$ Not an answer to your question but they have checked that the "first" order-of-trillions of zeroes appear on the line $\operatorname {Re}(z)=1/2$. I'm not sure how this compares to your version of heuristic proof. $\endgroup$ – Elliot G Dec 13 '16 at 18:21
  • $\begingroup$ It does, the lower values of $\zeta(s)$ are related to the real part of the lower zeros which we know are $1/2$ @ElliotG $\endgroup$ – reuns Dec 13 '16 at 18:22
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    $\begingroup$ @MathematicsStudent1122 And if the RH is false, what is the value ? $\endgroup$ – reuns Dec 13 '16 at 18:28
  • $\begingroup$ Already very few mathematicians seriously deny the validity of the conjecture because of numerical evidence... $\endgroup$ – Peter Humphries Dec 14 '16 at 18:56

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